LMFDB - Embedded newform 7220.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7220,2,Mod(1,7220)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7220, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7220 = 2^{2} \cdot 5 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7220.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$57.6519902594$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.133593.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 3x + 2 $ x^4 - x^3 - 8x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.16505$ of defining polynomial
Character	$\chi$	$=$	7220.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16505 q^{3} +1.00000 q^{5} -1.53315 q^{7} +7.01755 q^{9} -3.79695 q^{11} +6.69820 q^{13} -3.16505 q^{15} -6.01755 q^{17} +4.85250 q^{21} +4.38565 q^{23} +1.00000 q^{25} -12.7158 q^{27} +1.09875 q^{29} +1.05555 q^{31} +12.0175 q^{33} -1.53315 q^{35} +10.7970 q^{37} -21.2002 q^{39} -3.53315 q^{41} +5.55070 q^{43} +7.01755 q^{45} -11.2381 q^{47} -4.64945 q^{49} +19.0459 q^{51} -1.57636 q^{53} -3.79695 q^{55} -3.23135 q^{59} +0.0663000 q^{61} -10.7590 q^{63} +6.69820 q^{65} +5.70500 q^{67} -13.8808 q^{69} +6.47760 q^{71} +13.9796 q^{73} -3.16505 q^{75} +5.82130 q^{77} +1.99320 q^{79} +19.1934 q^{81} -10.3477 q^{83} -6.01755 q^{85} -3.47760 q^{87} -3.07310 q^{89} -10.2693 q^{91} -3.34086 q^{93} -8.87005 q^{97} -26.6453 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47}+ \cdots - 24 q^{99}+O(q^{100}) $ 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.16505	−1.82734	−0.913672	−	0.406453i	$-0.866765\pi$
$3$	−3.16505	−1.82734	−0.913672	+	0.406453i	$0.866765\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.53315	−0.579476	−0.289738	−	0.957106i	$-0.593568\pi$
$7$	−1.53315	−0.579476	−0.289738	+	0.957106i	$0.593568\pi$
$8$	0	0
$9$	7.01755	2.33918
$10$	0	0
$11$	−3.79695	−1.14482	−0.572412	−	0.819966i	$-0.693992\pi$
$11$	−3.79695	−1.14482	−0.572412	+	0.819966i	$0.693992\pi$
$12$	0	0
$13$	6.69820	1.85775	0.928873	−	0.370397i	$-0.120778\pi$
$13$	6.69820	1.85775	0.928873	+	0.370397i	$0.120778\pi$
$14$	0	0
$15$	−3.16505	−0.817213
$16$	0	0
$17$	−6.01755	−1.45947	−0.729735	−	0.683730i	$-0.760357\pi$
$17$	−6.01755	−1.45947	−0.729735	+	0.683730i	$0.760357\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.85250	1.05890
$22$	0	0
$23$	4.38565	0.914471	0.457235	−	0.889346i	$-0.348840\pi$
$23$	4.38565	0.914471	0.457235	+	0.889346i	$0.348840\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−12.7158	−2.44715
$28$	0	0
$29$	1.09875	0.204033	0.102016	−	0.994783i	$-0.467471\pi$
$29$	1.09875	0.204033	0.102016	+	0.994783i	$0.467471\pi$
$30$	0	0
$31$	1.05555	0.189582	0.0947908	−	0.995497i	$-0.469782\pi$
$31$	1.05555	0.189582	0.0947908	+	0.995497i	$0.469782\pi$
$32$	0	0
$33$	12.0175	2.09199
$34$	0	0
$35$	−1.53315	−0.259150
$36$	0	0
$37$	10.7970	1.77501	0.887504	−	0.460800i	$-0.152437\pi$
$37$	10.7970	1.77501	0.887504	+	0.460800i	$0.152437\pi$
$38$	0	0
$39$	−21.2002	−3.39474
$40$	0	0
$41$	−3.53315	−0.551785	−0.275893	−	0.961188i	$-0.588973\pi$
$41$	−3.53315	−0.551785	−0.275893	+	0.961188i	$0.588973\pi$
$42$	0	0
$43$	5.55070	0.846474	0.423237	−	0.906019i	$-0.360894\pi$
$43$	5.55070	0.846474	0.423237	+	0.906019i	$0.360894\pi$
$44$	0	0
$45$	7.01755	1.04611
$46$	0	0
$47$	−11.2381	−1.63925	−0.819626	−	0.572899i	$-0.805819\pi$
$47$	−11.2381	−1.63925	−0.819626	+	0.572899i	$0.805819\pi$
$48$	0	0
$49$	−4.64945	−0.664207
$50$	0	0
$51$	19.0459	2.66695
$52$	0	0
$53$	−1.57636	−0.216529	−0.108265	−	0.994122i	$-0.534529\pi$
$53$	−1.57636	−0.216529	−0.108265	+	0.994122i	$0.534529\pi$
$54$	0	0
$55$	−3.79695	−0.511981
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.23135	−0.420686	−0.210343	−	0.977628i	$-0.567458\pi$
$59$	−3.23135	−0.420686	−0.210343	+	0.977628i	$0.567458\pi$
$60$	0	0
$61$	0.0663000	0.00848885	0.00424442	−	0.999991i	$-0.498649\pi$
$61$	0.0663000	0.00848885	0.00424442	+	0.999991i	$0.498649\pi$
$62$	0	0
$63$	−10.7590	−1.35550
$64$	0	0
$65$	6.69820	0.830810
$66$	0	0
$67$	5.70500	0.696976	0.348488	−	0.937313i	$-0.386695\pi$
$67$	5.70500	0.696976	0.348488	+	0.937313i	$0.386695\pi$
$68$	0	0
$69$	−13.8808	−1.67105
$70$	0	0
$71$	6.47760	0.768750	0.384375	−	0.923177i	$-0.374417\pi$
$71$	6.47760	0.768750	0.384375	+	0.923177i	$0.374417\pi$
$72$	0	0
$73$	13.9796	1.63618	0.818091	−	0.575088i	$-0.195032\pi$
$73$	13.9796	1.63618	0.818091	+	0.575088i	$0.195032\pi$
$74$	0	0
$75$	−3.16505	−0.365469
$76$	0	0
$77$	5.82130	0.663398
$78$	0	0
$79$	1.99320	0.224253	0.112127	−	0.993694i	$-0.464234\pi$
$79$	1.99320	0.224253	0.112127	+	0.993694i	$0.464234\pi$
$80$	0	0
$81$	19.1934	2.13260
$82$	0	0
$83$	−10.3477	−1.13580	−0.567901	−	0.823097i	$-0.692244\pi$
$83$	−10.3477	−1.13580	−0.567901	+	0.823097i	$0.692244\pi$
$84$	0	0
$85$	−6.01755	−0.652695
$86$	0	0
$87$	−3.47760	−0.372838
$88$	0	0
$89$	−3.07310	−0.325747	−0.162874	−	0.986647i	$-0.552076\pi$
$89$	−3.07310	−0.325747	−0.162874	+	0.986647i	$0.552076\pi$
$90$	0	0
$91$	−10.2693	−1.07652
$92$	0	0
$93$	−3.34086	−0.346431
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−8.87005	−0.900617	−0.450308	−	0.892873i	$-0.648686\pi$
$97$	−8.87005	−0.900617	−0.450308	+	0.892873i	$0.648686\pi$
$98$	0	0
$99$	−26.6453	−2.67795
$100$	0	0
$101$	6.20826	0.617745	0.308872	−	0.951104i	$-0.400048\pi$
$101$	6.20826	0.617745	0.308872	+	0.951104i	$0.400048\pi$
$102$	0	0
$103$	0.253048	0.0249336	0.0124668	−	0.999922i	$-0.496032\pi$
$103$	0.253048	0.0249336	0.0124668	+	0.999922i	$0.496032\pi$
$104$	0	0
$105$	4.85250	0.473555
$106$	0	0
$107$	6.05555	0.585412	0.292706	−	0.956203i	$-0.405444\pi$
$107$	6.05555	0.585412	0.292706	+	0.956203i	$0.405444\pi$
$108$	0	0
$109$	−5.24890	−0.502754	−0.251377	−	0.967889i	$-0.580883\pi$
$109$	−5.24890	−0.502754	−0.251377	+	0.967889i	$0.580883\pi$
$110$	0	0
$111$	−34.1729	−3.24355
$112$	0	0
$113$	−2.42885	−0.228487	−0.114244	−	0.993453i	$-0.536444\pi$
$113$	−2.42885	−0.228487	−0.114244	+	0.993453i	$0.536444\pi$
$114$	0	0
$115$	4.38565	0.408964
$116$	0	0
$117$	47.0050	4.34561
$118$	0	0
$119$	9.22581	0.845728
$120$	0	0
$121$	3.41685	0.310623
$122$	0	0
$123$	11.1826	1.00830
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.2989	1.71250	0.856250	−	0.516561i	$-0.172788\pi$
$127$	19.2989	1.71250	0.856250	+	0.516561i	$0.172788\pi$
$128$	0	0
$129$	−17.5682	−1.54680
$130$	0	0
$131$	−13.2597	−1.15850	−0.579251	−	0.815149i	$-0.696655\pi$
$131$	−13.2597	−1.15850	−0.579251	+	0.815149i	$0.696655\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−12.7158	−1.09440
$136$	0	0
$137$	−15.5507	−1.32859	−0.664293	−	0.747472i	$-0.731267\pi$
$137$	−15.5507	−1.32859	−0.664293	+	0.747472i	$0.731267\pi$
$138$	0	0
$139$	−18.6222	−1.57952	−0.789758	−	0.613419i	$-0.789794\pi$
$139$	−18.6222	−1.57952	−0.789758	+	0.613419i	$0.789794\pi$
$140$	0	0
$141$	35.5693	2.99548
$142$	0	0
$143$	−25.4328	−2.12679
$144$	0	0
$145$	1.09875	0.0912463
$146$	0	0
$147$	14.7158	1.21373
$148$	0	0
$149$	−4.68745	−0.384011	−0.192005	−	0.981394i	$-0.561499\pi$
$149$	−4.68745	−0.384011	−0.192005	+	0.981394i	$0.561499\pi$
$150$	0	0
$151$	5.93370	0.482878	0.241439	−	0.970416i	$-0.422381\pi$
$151$	5.93370	0.482878	0.241439	+	0.970416i	$0.422381\pi$
$152$	0	0
$153$	−42.2285	−3.41397
$154$	0	0
$155$	1.05555	0.0847835
$156$	0	0
$157$	−9.29369	−0.741717	−0.370859	−	0.928689i	$-0.620937\pi$
$157$	−9.29369	−0.741717	−0.370859	+	0.928689i	$0.620937\pi$
$158$	0	0
$159$	4.98925	0.395673
$160$	0	0
$161$	−6.72386	−0.529914
$162$	0	0
$163$	5.88495	0.460945	0.230472	−	0.973079i	$-0.425973\pi$
$163$	5.88495	0.460945	0.230472	+	0.973079i	$0.425973\pi$
$164$	0	0
$165$	12.0175	0.935565
$166$	0	0
$167$	−12.7970	−0.990258	−0.495129	−	0.868819i	$-0.664879\pi$
$167$	−12.7970	−0.990258	−0.495129	+	0.868819i	$0.664879\pi$
$168$	0	0
$169$	31.8659	2.45122
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.0406	1.14352	0.571760	−	0.820421i	$-0.306261\pi$
$173$	15.0406	1.14352	0.571760	+	0.820421i	$0.306261\pi$
$174$	0	0
$175$	−1.53315	−0.115895
$176$	0	0
$177$	10.2274	0.768738
$178$	0	0
$179$	−21.5452	−1.61036	−0.805180	−	0.593030i	$-0.797931\pi$
$179$	−21.5452	−1.61036	−0.805180	+	0.593030i	$0.797931\pi$
$180$	0	0
$181$	−4.27060	−0.317431	−0.158716	−	0.987324i	$-0.550735\pi$
$181$	−4.27060	−0.317431	−0.158716	+	0.987324i	$0.550735\pi$
$182$	0	0
$183$	−0.209843	−0.0155120
$184$	0	0
$185$	10.7970	0.793808
$186$	0	0
$187$	22.8484	1.67084
$188$	0	0
$189$	19.4952	1.41806
$190$	0	0
$191$	−6.30180	−0.455982	−0.227991	−	0.973663i	$-0.573216\pi$
$191$	−6.30180	−0.455982	−0.227991	+	0.973663i	$0.573216\pi$
$192$	0	0
$193$	15.3789	1.10699	0.553497	−	0.832851i	$-0.313293\pi$
$193$	15.3789	1.10699	0.553497	+	0.832851i	$0.313293\pi$
$194$	0	0
$195$	−21.2002	−1.51817
$196$	0	0
$197$	−12.7210	−0.906331	−0.453165	−	0.891426i	$-0.649705\pi$
$197$	−12.7210	−0.906331	−0.453165	+	0.891426i	$0.649705\pi$
$198$	0	0
$199$	10.2177	0.724314	0.362157	−	0.932117i	$-0.382040\pi$
$199$	10.2177	0.724314	0.362157	+	0.932117i	$0.382040\pi$
$200$	0	0
$201$	−18.0566	−1.27361
$202$	0	0
$203$	−1.68455	−0.118232
$204$	0	0
$205$	−3.53315	−0.246766
$206$	0	0
$207$	30.7765	2.13912
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−5.07310	−0.349246	−0.174623	−	0.984635i	$-0.555871\pi$
$211$	−5.07310	−0.349246	−0.174623	+	0.984635i	$0.555871\pi$
$212$	0	0
$213$	−20.5019	−1.40477
$214$	0	0
$215$	5.55070	0.378555
$216$	0	0
$217$	−1.61831	−0.109858
$218$	0	0
$219$	−44.2460	−2.98987
$220$	0	0
$221$	−40.3068	−2.71133
$222$	0	0
$223$	8.52635	0.570967	0.285483	−	0.958384i	$-0.407846\pi$
$223$	8.52635	0.570967	0.285483	+	0.958384i	$0.407846\pi$
$224$	0	0
$225$	7.01755	0.467837
$226$	0	0
$227$	20.0566	1.33120	0.665602	−	0.746307i	$-0.268175\pi$
$227$	20.0566	1.33120	0.665602	+	0.746307i	$0.268175\pi$
$228$	0	0
$229$	7.92955	0.524000	0.262000	−	0.965068i	$-0.415618\pi$
$229$	7.92955	0.524000	0.262000	+	0.965068i	$0.415618\pi$
$230$	0	0
$231$	−18.4247	−1.21226
$232$	0	0
$233$	10.5790	0.693054	0.346527	−	0.938040i	$-0.387361\pi$
$233$	10.5790	0.693054	0.346527	+	0.938040i	$0.387361\pi$
$234$	0	0
$235$	−11.2381	−0.733096
$236$	0	0
$237$	−6.30859	−0.409787
$238$	0	0
$239$	−5.68065	−0.367451	−0.183725	−	0.982978i	$-0.558816\pi$
$239$	−5.68065	−0.367451	−0.183725	+	0.982978i	$0.558816\pi$
$240$	0	0
$241$	−9.51429	−0.612869	−0.306435	−	0.951892i	$-0.599136\pi$
$241$	−9.51429	−0.612869	−0.306435	+	0.951892i	$0.599136\pi$
$242$	0	0
$243$	−22.6007	−1.44984
$244$	0	0
$245$	−4.64945	−0.297043
$246$	0	0
$247$	0	0
$248$	0	0
$249$	32.7509	2.07550
$250$	0	0
$251$	12.5332	0.791085	0.395543	−	0.918448i	$-0.370557\pi$
$251$	12.5332	0.791085	0.395543	+	0.918448i	$0.370557\pi$
$252$	0	0
$253$	−16.6521	−1.04691
$254$	0	0
$255$	19.0459	1.19270
$256$	0	0
$257$	17.0891	1.06599	0.532993	−	0.846120i	$-0.321067\pi$
$257$	17.0891	1.06599	0.532993	+	0.846120i	$0.321067\pi$
$258$	0	0
$259$	−16.5533	−1.02857
$260$	0	0
$261$	7.71054	0.477271
$262$	0	0
$263$	0.920109	0.0567363	0.0283682	−	0.999598i	$-0.490969\pi$
$263$	0.920109	0.0567363	0.0283682	+	0.999598i	$0.490969\pi$
$264$	0	0
$265$	−1.57636	−0.0968347
$266$	0	0
$267$	9.72651	0.595252
$268$	0	0
$269$	−25.3652	−1.54654	−0.773272	−	0.634075i	$-0.781381\pi$
$269$	−25.3652	−1.54654	−0.773272	+	0.634075i	$0.781381\pi$
$270$	0	0
$271$	−11.2366	−0.682572	−0.341286	−	0.939959i	$-0.610863\pi$
$271$	−11.2366	−0.682572	−0.341286	+	0.939959i	$0.610863\pi$
$272$	0	0
$273$	32.5030	1.96717
$274$	0	0
$275$	−3.79695	−0.228965
$276$	0	0
$277$	16.5491	0.994340	0.497170	−	0.867653i	$-0.334373\pi$
$277$	16.5491	0.994340	0.497170	+	0.867653i	$0.334373\pi$
$278$	0	0
$279$	7.40735	0.443466
$280$	0	0
$281$	10.5790	0.631090	0.315545	−	0.948911i	$-0.397813\pi$
$281$	10.5790	0.631090	0.315545	+	0.948911i	$0.397813\pi$
$282$	0	0
$283$	20.7414	1.23295	0.616474	−	0.787375i	$-0.288560\pi$
$283$	20.7414	1.23295	0.616474	+	0.787375i	$0.288560\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.41685	0.319746
$288$	0	0
$289$	19.2109	1.13005
$290$	0	0
$291$	28.0742	1.64574
$292$	0	0
$293$	−13.6846	−0.799460	−0.399730	−	0.916633i	$-0.630896\pi$
$293$	−13.6846	−0.799460	−0.399730	+	0.916633i	$0.630896\pi$
$294$	0	0
$295$	−3.23135	−0.188137
$296$	0	0
$297$	48.2811	2.80155
$298$	0	0
$299$	29.3760	1.69886
$300$	0	0
$301$	−8.51006	−0.490511
$302$	0	0
$303$	−19.6495	−1.12883
$304$	0	0
$305$	0.0663000	0.00379633
$306$	0	0
$307$	8.63745	0.492965	0.246483	−	0.969147i	$-0.420725\pi$
$307$	8.63745	0.492965	0.246483	+	0.969147i	$0.420725\pi$
$308$	0	0
$309$	−0.800911	−0.0455622
$310$	0	0
$311$	29.7658	1.68786	0.843930	−	0.536453i	$-0.180236\pi$
$311$	29.7658	1.68786	0.843930	+	0.536453i	$0.180236\pi$
$312$	0	0
$313$	11.2083	0.633528	0.316764	−	0.948504i	$-0.397404\pi$
$313$	11.2083	0.633528	0.316764	+	0.948504i	$0.397404\pi$
$314$	0	0
$315$	−10.7590	−0.606199
$316$	0	0
$317$	4.69299	0.263585	0.131792	−	0.991277i	$-0.457927\pi$
$317$	4.69299	0.263585	0.131792	+	0.991277i	$0.457927\pi$
$318$	0	0
$319$	−4.17191	−0.233582
$320$	0	0
$321$	−19.1661	−1.06975
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.69820	0.371549
$326$	0	0
$327$	16.6130	0.918703
$328$	0	0
$329$	17.2298	0.949908
$330$	0	0
$331$	−29.1137	−1.60024	−0.800118	−	0.599843i	$-0.795230\pi$
$331$	−29.1137	−1.60024	−0.800118	+	0.599843i	$0.795230\pi$
$332$	0	0
$333$	75.7682	4.15207
$334$	0	0
$335$	5.70500	0.311697
$336$	0	0
$337$	−23.1580	−1.26150	−0.630749	−	0.775987i	$-0.717252\pi$
$337$	−23.1580	−1.26150	−0.630749	+	0.775987i	$0.717252\pi$
$338$	0	0
$339$	7.68745	0.417525
$340$	0	0
$341$	−4.00786	−0.217038
$342$	0	0
$343$	17.8604	0.964369
$344$	0	0
$345$	−13.8808	−0.747317
$346$	0	0
$347$	15.6618	0.840769	0.420385	−	0.907346i	$-0.361895\pi$
$347$	15.6618	0.840769	0.420385	+	0.907346i	$0.361895\pi$
$348$	0	0
$349$	−9.72130	−0.520369	−0.260185	−	0.965559i	$-0.583783\pi$
$349$	−9.72130	−0.520369	−0.260185	+	0.965559i	$0.583783\pi$
$350$	0	0
$351$	−85.1727	−4.54618
$352$	0	0
$353$	11.3789	0.605635	0.302818	−	0.953049i	$-0.402073\pi$
$353$	11.3789	0.605635	0.302818	+	0.953049i	$0.402073\pi$
$354$	0	0
$355$	6.47760	0.343796
$356$	0	0
$357$	−29.2002	−1.54544
$358$	0	0
$359$	28.7753	1.51870	0.759350	−	0.650682i	$-0.225517\pi$
$359$	28.7753	1.51870	0.759350	+	0.650682i	$0.225517\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−10.8145	−0.567614
$364$	0	0
$365$	13.9796	0.731723
$366$	0	0
$367$	−24.1297	−1.25956	−0.629780	−	0.776773i	$-0.716855\pi$
$367$	−24.1297	−1.25956	−0.629780	+	0.776773i	$0.716855\pi$
$368$	0	0
$369$	−24.7941	−1.29073
$370$	0	0
$371$	2.41679	0.125473
$372$	0	0
$373$	26.7281	1.38393	0.691964	−	0.721932i	$-0.256746\pi$
$373$	26.7281	1.38393	0.691964	+	0.721932i	$0.256746\pi$
$374$	0	0
$375$	−3.16505	−0.163443
$376$	0	0
$377$	7.35966	0.379042
$378$	0	0
$379$	−4.29369	−0.220552	−0.110276	−	0.993901i	$-0.535174\pi$
$379$	−4.29369	−0.220552	−0.110276	+	0.993901i	$0.535174\pi$
$380$	0	0
$381$	−61.0820	−3.12933
$382$	0	0
$383$	−5.57221	−0.284727	−0.142363	−	0.989814i	$-0.545470\pi$
$383$	−5.57221	−0.284727	−0.142363	+	0.989814i	$0.545470\pi$
$384$	0	0
$385$	5.82130	0.296681
$386$	0	0
$387$	38.9523	1.98006
$388$	0	0
$389$	28.1486	1.42719	0.713594	−	0.700559i	$-0.247066\pi$
$389$	28.1486	1.42719	0.713594	+	0.700559i	$0.247066\pi$
$390$	0	0
$391$	−26.3909	−1.33464
$392$	0	0
$393$	41.9675	2.11698
$394$	0	0
$395$	1.99320	0.100289
$396$	0	0
$397$	34.7425	1.74367	0.871837	−	0.489796i	$-0.162929\pi$
$397$	34.7425	1.74367	0.871837	+	0.489796i	$0.162929\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	23.8740	1.19221	0.596106	−	0.802906i	$-0.296714\pi$
$401$	23.8740	1.19221	0.596106	+	0.802906i	$0.296714\pi$
$402$	0	0
$403$	7.07026	0.352195
$404$	0	0
$405$	19.1934	0.953725
$406$	0	0
$407$	−40.9955	−2.03207
$408$	0	0
$409$	37.4692	1.85273	0.926365	−	0.376626i	$-0.122916\pi$
$409$	37.4692	1.85273	0.926365	+	0.376626i	$0.122916\pi$
$410$	0	0
$411$	49.2188	2.42778
$412$	0	0
$413$	4.95415	0.243778
$414$	0	0
$415$	−10.3477	−0.507946
$416$	0	0
$417$	58.9402	2.88632
$418$	0	0
$419$	−0.360241	−0.0175989	−0.00879947	−	0.999961i	$-0.502801\pi$
$419$	−0.360241	−0.0175989	−0.00879947	+	0.999961i	$0.502801\pi$
$420$	0	0
$421$	−24.2486	−1.18180	−0.590901	−	0.806744i	$-0.701228\pi$
$421$	−24.2486	−1.18180	−0.590901	+	0.806744i	$0.701228\pi$
$422$	0	0
$423$	−78.8643	−3.83451
$424$	0	0
$425$	−6.01755	−0.291894
$426$	0	0
$427$	−0.101648	−0.00491909
$428$	0	0
$429$	80.4960	3.88638
$430$	0	0
$431$	12.6413	0.608912	0.304456	−	0.952526i	$-0.401525\pi$
$431$	12.6413	0.608912	0.304456	+	0.952526i	$0.401525\pi$
$432$	0	0
$433$	24.4640	1.17566	0.587831	−	0.808984i	$-0.299982\pi$
$433$	24.4640	1.17566	0.587831	+	0.808984i	$0.299982\pi$
$434$	0	0
$435$	−3.47760	−0.166738
$436$	0	0
$437$	0	0
$438$	0	0
$439$	24.8009	1.18368	0.591840	−	0.806055i	$-0.298402\pi$
$439$	24.8009	1.18368	0.591840	+	0.806055i	$0.298402\pi$
$440$	0	0
$441$	−32.6278	−1.55370
$442$	0	0
$443$	30.2636	1.43786	0.718932	−	0.695080i	$-0.244631\pi$
$443$	30.2636	1.43786	0.718932	+	0.695080i	$0.244631\pi$
$444$	0	0
$445$	−3.07310	−0.145679
$446$	0	0
$447$	14.8360	0.701719
$448$	0	0
$449$	29.0742	1.37209	0.686047	−	0.727557i	$-0.259344\pi$
$449$	29.0742	1.37209	0.686047	+	0.727557i	$0.259344\pi$
$450$	0	0
$451$	13.4152	0.631697
$452$	0	0
$453$	−18.7805	−0.882383
$454$	0	0
$455$	−10.2693	−0.481434
$456$	0	0
$457$	−3.06234	−0.143250	−0.0716251	−	0.997432i	$-0.522819\pi$
$457$	−3.06234	−0.143250	−0.0716251	+	0.997432i	$0.522819\pi$
$458$	0	0
$459$	76.5177	3.57154
$460$	0	0
$461$	−26.5869	−1.23827	−0.619137	−	0.785283i	$-0.712517\pi$
$461$	−26.5869	−1.23827	−0.619137	+	0.785283i	$0.712517\pi$
$462$	0	0
$463$	4.82684	0.224322	0.112161	−	0.993690i	$-0.464223\pi$
$463$	4.82684	0.224322	0.112161	+	0.993690i	$0.464223\pi$
$464$	0	0
$465$	−3.34086	−0.154929
$466$	0	0
$467$	−1.31539	−0.0608690	−0.0304345	−	0.999537i	$-0.509689\pi$
$467$	−1.31539	−0.0608690	−0.0304345	+	0.999537i	$0.509689\pi$
$468$	0	0
$469$	−8.74662	−0.403881
$470$	0	0
$471$	29.4150	1.35537
$472$	0	0
$473$	−21.0757	−0.969064
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−11.0622	−0.506501
$478$	0	0
$479$	19.0783	0.871710	0.435855	−	0.900017i	$-0.356446\pi$
$479$	19.0783	0.871710	0.435855	+	0.900017i	$0.356446\pi$
$480$	0	0
$481$	72.3202	3.29752
$482$	0	0
$483$	21.2814	0.968335
$484$	0	0
$485$	−8.87005	−0.402768
$486$	0	0
$487$	8.23656	0.373234	0.186617	−	0.982433i	$-0.440248\pi$
$487$	8.23656	0.373234	0.186617	+	0.982433i	$0.440248\pi$
$488$	0	0
$489$	−18.6262	−0.842304
$490$	0	0
$491$	30.9551	1.39699	0.698493	−	0.715617i	$-0.253854\pi$
$491$	30.9551	1.39699	0.698493	+	0.715617i	$0.253854\pi$
$492$	0	0
$493$	−6.61179	−0.297780
$494$	0	0
$495$	−26.6453	−1.19762
$496$	0	0
$497$	−9.93114	−0.445472
$498$	0	0
$499$	1.10951	0.0496683	0.0248341	−	0.999692i	$-0.492094\pi$
$499$	1.10951	0.0496683	0.0248341	+	0.999692i	$0.492094\pi$
$500$	0	0
$501$	40.5030	1.80954
$502$	0	0
$503$	8.65104	0.385731	0.192865	−	0.981225i	$-0.438222\pi$
$503$	8.65104	0.385731	0.192865	+	0.981225i	$0.438222\pi$
$504$	0	0
$505$	6.20826	0.276264
$506$	0	0
$507$	−100.857	−4.47923
$508$	0	0
$509$	20.7888	0.921449	0.460725	−	0.887543i	$-0.347590\pi$
$509$	20.7888	0.921449	0.460725	+	0.887543i	$0.347590\pi$
$510$	0	0
$511$	−21.4328	−0.948129
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.253048	0.0111506
$516$	0	0
$517$	42.6707	1.87666
$518$	0	0
$519$	−47.6044	−2.08960
$520$	0	0
$521$	24.5098	1.07379	0.536897	−	0.843648i	$-0.319596\pi$
$521$	24.5098	1.07379	0.536897	+	0.843648i	$0.319596\pi$
$522$	0	0
$523$	−18.0689	−0.790100	−0.395050	−	0.918660i	$-0.629273\pi$
$523$	−18.0689	−0.790100	−0.395050	+	0.918660i	$0.629273\pi$
$524$	0	0
$525$	4.85250	0.211780
$526$	0	0
$527$	−6.35180	−0.276689
$528$	0	0
$529$	−3.76609	−0.163743
$530$	0	0
$531$	−22.6762	−0.984062
$532$	0	0
$533$	−23.6658	−1.02508
$534$	0	0
$535$	6.05555	0.261804
$536$	0	0
$537$	68.1915	2.94268
$538$	0	0
$539$	17.6537	0.760401
$540$	0	0
$541$	−17.5088	−0.752762	−0.376381	−	0.926465i	$-0.622832\pi$
$541$	−17.5088	−0.752762	−0.376381	+	0.926465i	$0.622832\pi$
$542$	0	0
$543$	13.5167	0.580055
$544$	0	0
$545$	−5.24890	−0.224838
$546$	0	0
$547$	−26.9728	−1.15327	−0.576636	−	0.817001i	$-0.695635\pi$
$547$	−26.9728	−1.15327	−0.576636	+	0.817001i	$0.695635\pi$
$548$	0	0
$549$	0.465264	0.0198570
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−3.05588	−0.129949
$554$	0	0
$555$	−34.1729	−1.45056
$556$	0	0
$557$	−15.9672	−0.676553	−0.338276	−	0.941047i	$-0.609844\pi$
$557$	−15.9672	−0.676553	−0.338276	+	0.941047i	$0.609844\pi$
$558$	0	0
$559$	37.1797	1.57253
$560$	0	0
$561$	−72.3162	−3.05319
$562$	0	0
$563$	8.58711	0.361904	0.180952	−	0.983492i	$-0.442082\pi$
$563$	8.58711	0.361904	0.180952	+	0.983492i	$0.442082\pi$
$564$	0	0
$565$	−2.42885	−0.102183
$566$	0	0
$567$	−29.4263	−1.23579
$568$	0	0
$569$	−22.5910	−0.947064	−0.473532	−	0.880777i	$-0.657021\pi$
$569$	−22.5910	−0.947064	−0.473532	+	0.880777i	$0.657021\pi$
$570$	0	0
$571$	17.1000	0.715613	0.357806	−	0.933796i	$-0.383525\pi$
$571$	17.1000	0.715613	0.357806	+	0.933796i	$0.383525\pi$
$572$	0	0
$573$	19.9455	0.833236
$574$	0	0
$575$	4.38565	0.182894
$576$	0	0
$577$	−0.677755	−0.0282153	−0.0141077	−	0.999900i	$-0.504491\pi$
$577$	−0.677755	−0.0282153	−0.0141077	+	0.999900i	$0.504491\pi$
$578$	0	0
$579$	−48.6749	−2.02286
$580$	0	0
$581$	15.8645	0.658171
$582$	0	0
$583$	5.98535	0.247888
$584$	0	0
$585$	47.0050	1.94342
$586$	0	0
$587$	27.3718	1.12976	0.564878	−	0.825175i	$-0.308923\pi$
$587$	27.3718	1.12976	0.564878	+	0.825175i	$0.308923\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	40.2625	1.65618
$592$	0	0
$593$	8.37879	0.344076	0.172038	−	0.985090i	$-0.444965\pi$
$593$	8.37879	0.344076	0.172038	+	0.985090i	$0.444965\pi$
$594$	0	0
$595$	9.22581	0.378221
$596$	0	0
$597$	−32.3395	−1.32357
$598$	0	0
$599$	33.9107	1.38555	0.692777	−	0.721152i	$-0.256387\pi$
$599$	33.9107	1.38555	0.692777	+	0.721152i	$0.256387\pi$
$600$	0	0
$601$	5.74951	0.234528	0.117264	−	0.993101i	$-0.462588\pi$
$601$	5.74951	0.234528	0.117264	+	0.993101i	$0.462588\pi$
$602$	0	0
$603$	40.0351	1.63036
$604$	0	0
$605$	3.41685	0.138915
$606$	0	0
$607$	8.87815	0.360353	0.180177	−	0.983634i	$-0.442333\pi$
$607$	8.87815	0.360353	0.180177	+	0.983634i	$0.442333\pi$
$608$	0	0
$609$	5.33169	0.216051
$610$	0	0
$611$	−75.2754	−3.04532
$612$	0	0
$613$	−12.0420	−0.486370	−0.243185	−	0.969980i	$-0.578192\pi$
$613$	−12.0420	−0.486370	−0.243185	+	0.969980i	$0.578192\pi$
$614$	0	0
$615$	11.1826	0.450926
$616$	0	0
$617$	46.8724	1.88701	0.943505	−	0.331358i	$-0.107507\pi$
$617$	46.8724	1.88701	0.943505	+	0.331358i	$0.107507\pi$
$618$	0	0
$619$	14.8700	0.597678	0.298839	−	0.954304i	$-0.403401\pi$
$619$	14.8700	0.597678	0.298839	+	0.954304i	$0.403401\pi$
$620$	0	0
$621$	−55.7668	−2.23785
$622$	0	0
$623$	4.71152	0.188763
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−64.9712	−2.59057
$630$	0	0
$631$	17.4247	0.693667	0.346833	−	0.937927i	$-0.387257\pi$
$631$	17.4247	0.693667	0.346833	+	0.937927i	$0.387257\pi$
$632$	0	0
$633$	16.0566	0.638193
$634$	0	0
$635$	19.2989	0.765854
$636$	0	0
$637$	−31.1430	−1.23393
$638$	0	0
$639$	45.4569	1.79825
$640$	0	0
$641$	−31.9945	−1.26371	−0.631854	−	0.775087i	$-0.717706\pi$
$641$	−31.9945	−1.26371	−0.631854	+	0.775087i	$0.717706\pi$
$642$	0	0
$643$	35.0406	1.38187	0.690934	−	0.722918i	$-0.257199\pi$
$643$	35.0406	1.38187	0.690934	+	0.722918i	$0.257199\pi$
$644$	0	0
$645$	−17.5682	−0.691749
$646$	0	0
$647$	17.4260	0.685087	0.342544	−	0.939502i	$-0.388712\pi$
$647$	17.4260	0.685087	0.342544	+	0.939502i	$0.388712\pi$
$648$	0	0
$649$	12.2693	0.481612
$650$	0	0
$651$	5.12203	0.200748
$652$	0	0
$653$	−49.1457	−1.92322	−0.961609	−	0.274422i	$-0.911513\pi$
$653$	−49.1457	−1.92322	−0.961609	+	0.274422i	$0.911513\pi$
$654$	0	0
$655$	−13.2597	−0.518098
$656$	0	0
$657$	98.1022	3.82733
$658$	0	0
$659$	−1.72544	−0.0672137	−0.0336069	−	0.999435i	$-0.510699\pi$
$659$	−1.72544	−0.0672137	−0.0336069	+	0.999435i	$0.510699\pi$
$660$	0	0
$661$	−21.9187	−0.852540	−0.426270	−	0.904596i	$-0.640173\pi$
$661$	−21.9187	−0.852540	−0.426270	+	0.904596i	$0.640173\pi$
$662$	0	0
$663$	127.573	4.95452
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.81874	0.186582
$668$	0	0
$669$	−26.9863	−1.04335
$670$	0	0
$671$	−0.251738	−0.00971824
$672$	0	0
$673$	−6.63165	−0.255631	−0.127816	−	0.991798i	$-0.540797\pi$
$673$	−6.63165	−0.255631	−0.127816	+	0.991798i	$0.540797\pi$
$674$	0	0
$675$	−12.7158	−0.489429
$676$	0	0
$677$	−32.7008	−1.25679	−0.628397	−	0.777893i	$-0.716289\pi$
$677$	−32.7008	−1.25679	−0.628397	+	0.777893i	$0.716289\pi$
$678$	0	0
$679$	13.5991	0.521886
$680$	0	0
$681$	−63.4802	−2.43257
$682$	0	0
$683$	51.9872	1.98923	0.994617	−	0.103622i	$-0.0330432\pi$
$683$	51.9872	1.98923	0.994617	+	0.103622i	$0.0330432\pi$
$684$	0	0
$685$	−15.5507	−0.594162
$686$	0	0
$687$	−25.0974	−0.957527
$688$	0	0
$689$	−10.5587	−0.402256
$690$	0	0
$691$	−8.93635	−0.339955	−0.169977	−	0.985448i	$-0.554369\pi$
$691$	−8.93635	−0.339955	−0.169977	+	0.985448i	$0.554369\pi$
$692$	0	0
$693$	40.8512	1.55181
$694$	0	0
$695$	−18.6222	−0.706381
$696$	0	0
$697$	21.2609	0.805314
$698$	0	0
$699$	−33.4831	−1.26645
$700$	0	0
$701$	2.25853	0.0853036	0.0426518	−	0.999090i	$-0.486419\pi$
$701$	2.25853	0.0853036	0.0426518	+	0.999090i	$0.486419\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	35.5693	1.33962
$706$	0	0
$707$	−9.51819	−0.357968
$708$	0	0
$709$	46.8090	1.75795	0.878975	−	0.476867i	$-0.158228\pi$
$709$	46.8090	1.75795	0.878975	+	0.476867i	$0.158228\pi$
$710$	0	0
$711$	13.9874	0.524569
$712$	0	0
$713$	4.62925	0.173367
$714$	0	0
$715$	−25.4328	−0.951131
$716$	0	0
$717$	17.9796	0.671459
$718$	0	0
$719$	−23.5140	−0.876925	−0.438462	−	0.898750i	$-0.644477\pi$
$719$	−23.5140	−0.876925	−0.438462	+	0.898750i	$0.644477\pi$
$720$	0	0
$721$	−0.387961	−0.0144484
$722$	0	0
$723$	30.1132	1.11992
$724$	0	0
$725$	1.09875	0.0408066
$726$	0	0
$727$	−25.4747	−0.944805	−0.472402	−	0.881383i	$-0.656613\pi$
$727$	−25.4747	−0.944805	−0.472402	+	0.881383i	$0.656613\pi$
$728$	0	0
$729$	13.9523	0.516752
$730$	0	0
$731$	−33.4016	−1.23540
$732$	0	0
$733$	−1.41916	−0.0524179	−0.0262090	−	0.999656i	$-0.508344\pi$
$733$	−1.41916	−0.0524179	−0.0262090	+	0.999656i	$0.508344\pi$
$734$	0	0
$735$	14.7158	0.542799
$736$	0	0
$737$	−21.6616	−0.797915
$738$	0	0
$739$	−21.5398	−0.792353	−0.396176	−	0.918174i	$-0.629663\pi$
$739$	−21.5398	−0.792353	−0.396176	+	0.918174i	$0.629663\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	42.5625	1.56147	0.780734	−	0.624864i	$-0.214846\pi$
$743$	42.5625	1.56147	0.780734	+	0.624864i	$0.214846\pi$
$744$	0	0
$745$	−4.68745	−0.171735
$746$	0	0
$747$	−72.6152	−2.65685
$748$	0	0
$749$	−9.28406	−0.339232
$750$	0	0
$751$	−18.6142	−0.679240	−0.339620	−	0.940563i	$-0.610299\pi$
$751$	−18.6142	−0.679240	−0.339620	+	0.940563i	$0.610299\pi$
$752$	0	0
$753$	−39.6681	−1.44558
$754$	0	0
$755$	5.93370	0.215949
$756$	0	0
$757$	−30.7684	−1.11830	−0.559148	−	0.829068i	$-0.688872\pi$
$757$	−30.7684	−1.11830	−0.559148	+	0.829068i	$0.688872\pi$
$758$	0	0
$759$	52.7047	1.91306
$760$	0	0
$761$	13.9270	0.504853	0.252427	−	0.967616i	$-0.418771\pi$
$761$	13.9270	0.504853	0.252427	+	0.967616i	$0.418771\pi$
$762$	0	0
$763$	8.04735	0.291334
$764$	0	0
$765$	−42.2285	−1.52677
$766$	0	0
$767$	−21.6442	−0.781528
$768$	0	0
$769$	−48.6332	−1.75376	−0.876880	−	0.480710i	$-0.840379\pi$
$769$	−48.6332	−1.75376	−0.876880	+	0.480710i	$0.840379\pi$
$770$	0	0
$771$	−54.0877	−1.94792
$772$	0	0
$773$	19.5855	0.704442	0.352221	−	0.935917i	$-0.385426\pi$
$773$	19.5855	0.704442	0.352221	+	0.935917i	$0.385426\pi$
$774$	0	0
$775$	1.05555	0.0379163
$776$	0	0
$777$	52.3922	1.87956
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−24.5952	−0.880084
$782$	0	0
$783$	−13.9714	−0.499299
$784$	0	0
$785$	−9.29369	−0.331706
$786$	0	0
$787$	−13.4029	−0.477760	−0.238880	−	0.971049i	$-0.576780\pi$
$787$	−13.4029	−0.477760	−0.238880	+	0.971049i	$0.576780\pi$
$788$	0	0
$789$	−2.91219	−0.103677
$790$	0	0
$791$	3.72380	0.132403
$792$	0	0
$793$	0.444091	0.0157701
$794$	0	0
$795$	4.98925	0.176950
$796$	0	0
$797$	−30.0201	−1.06337	−0.531684	−	0.846943i	$-0.678441\pi$
$797$	−30.0201	−1.06337	−0.531684	+	0.846943i	$0.678441\pi$
$798$	0	0
$799$	67.6261	2.39244
$800$	0	0
$801$	−21.5656	−0.761983
$802$	0	0
$803$	−53.0797	−1.87314
$804$	0	0
$805$	−6.72386	−0.236985
$806$	0	0
$807$	80.2822	2.82607
$808$	0	0
$809$	−13.4967	−0.474520	−0.237260	−	0.971446i	$-0.576249\pi$
$809$	−13.4967	−0.474520	−0.237260	+	0.971446i	$0.576249\pi$
$810$	0	0
$811$	5.26520	0.184886	0.0924431	−	0.995718i	$-0.470532\pi$
$811$	5.26520	0.184886	0.0924431	+	0.995718i	$0.470532\pi$
$812$	0	0
$813$	35.5643	1.24729
$814$	0	0
$815$	5.88495	0.206141
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−72.0657	−2.51818
$820$	0	0
$821$	−32.8295	−1.14576	−0.572879	−	0.819640i	$-0.694173\pi$
$821$	−32.8295	−1.14576	−0.572879	+	0.819640i	$0.694173\pi$
$822$	0	0
$823$	41.7154	1.45411	0.727054	−	0.686580i	$-0.240889\pi$
$823$	41.7154	1.45411	0.727054	+	0.686580i	$0.240889\pi$
$824$	0	0
$825$	12.0175	0.418397
$826$	0	0
$827$	12.1905	0.423904	0.211952	−	0.977280i	$-0.432018\pi$
$827$	12.1905	0.423904	0.211952	+	0.977280i	$0.432018\pi$
$828$	0	0
$829$	52.9958	1.84062	0.920310	−	0.391190i	$-0.127936\pi$
$829$	52.9958	1.84062	0.920310	+	0.391190i	$0.127936\pi$
$830$	0	0
$831$	−52.3788	−1.81700
$832$	0	0
$833$	27.9783	0.969391
$834$	0	0
$835$	−12.7970	−0.442857
$836$	0	0
$837$	−13.4221	−0.463934
$838$	0	0
$839$	15.2164	0.525330	0.262665	−	0.964887i	$-0.415399\pi$
$839$	15.2164	0.525330	0.262665	+	0.964887i	$0.415399\pi$
$840$	0	0
$841$	−27.7927	−0.958371
$842$	0	0
$843$	−33.4831	−1.15322
$844$	0	0
$845$	31.8659	1.09622
$846$	0	0
$847$	−5.23854	−0.179998
$848$	0	0
$849$	−65.6476	−2.25302
$850$	0	0
$851$	47.3516	1.62319
$852$	0	0
$853$	−47.1459	−1.61424	−0.807122	−	0.590385i	$-0.798976\pi$
$853$	−47.1459	−1.61424	−0.807122	+	0.590385i	$0.798976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.3812	1.75515	0.877574	−	0.479442i	$-0.159161\pi$
$857$	51.3812	1.75515	0.877574	+	0.479442i	$0.159161\pi$
$858$	0	0
$859$	2.06109	0.0703235	0.0351618	−	0.999382i	$-0.488805\pi$
$859$	2.06109	0.0703235	0.0351618	+	0.999382i	$0.488805\pi$
$860$	0	0
$861$	−17.1446	−0.584287
$862$	0	0
$863$	35.8646	1.22084	0.610422	−	0.792076i	$-0.291000\pi$
$863$	35.8646	1.22084	0.610422	+	0.792076i	$0.291000\pi$
$864$	0	0
$865$	15.0406	0.511397
$866$	0	0
$867$	−60.8035	−2.06500
$868$	0	0
$869$	−7.56810	−0.256730
$870$	0	0
$871$	38.2132	1.29481
$872$	0	0
$873$	−62.2460	−2.10671
$874$	0	0
$875$	−1.53315	−0.0518299
$876$	0	0
$877$	−13.0920	−0.442084	−0.221042	−	0.975264i	$-0.570946\pi$
$877$	−13.0920	−0.442084	−0.221042	+	0.975264i	$0.570946\pi$
$878$	0	0
$879$	43.3123	1.46089
$880$	0	0
$881$	−26.9322	−0.907369	−0.453684	−	0.891162i	$-0.649891\pi$
$881$	−26.9322	−0.907369	−0.453684	+	0.891162i	$0.649891\pi$
$882$	0	0
$883$	−29.6345	−0.997280	−0.498640	−	0.866809i	$-0.666167\pi$
$883$	−29.6345	−0.997280	−0.498640	+	0.866809i	$0.666167\pi$
$884$	0	0
$885$	10.2274	0.343790
$886$	0	0
$887$	30.4302	1.02175	0.510873	−	0.859656i	$-0.329322\pi$
$887$	30.4302	1.02175	0.510873	+	0.859656i	$0.329322\pi$
$888$	0	0
$889$	−29.5881	−0.992353
$890$	0	0
$891$	−72.8763	−2.44145
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−21.5452	−0.720175
$896$	0	0
$897$	−92.9764	−3.10439
$898$	0	0
$899$	1.15978	0.0386809
$900$	0	0
$901$	9.48580	0.316018
$902$	0	0
$903$	26.9348	0.896333
$904$	0	0
$905$	−4.27060	−0.141959
$906$	0	0
$907$	−11.1936	−0.371678	−0.185839	−	0.982580i	$-0.559500\pi$
$907$	−11.1936	−0.371678	−0.185839	+	0.982580i	$0.559500\pi$
$908$	0	0
$909$	43.5668	1.44502
$910$	0	0
$911$	0.339795	0.0112579	0.00562895	−	0.999984i	$-0.498208\pi$
$911$	0.339795	0.0112579	0.00562895	+	0.999984i	$0.498208\pi$
$912$	0	0
$913$	39.2895	1.30029
$914$	0	0
$915$	−0.209843	−0.00693720
$916$	0	0
$917$	20.3290	0.671324
$918$	0	0
$919$	−15.0715	−0.497163	−0.248582	−	0.968611i	$-0.579964\pi$
$919$	−15.0715	−0.497163	−0.248582	+	0.968611i	$0.579964\pi$
$920$	0	0
$921$	−27.3380	−0.900816
$922$	0	0
$923$	43.3883	1.42814
$924$	0	0
$925$	10.7970	0.355002
$926$	0	0
$927$	1.77578	0.0583243
$928$	0	0
$929$	22.6557	0.743310	0.371655	−	0.928371i	$-0.378790\pi$
$929$	22.6557	0.743310	0.371655	+	0.928371i	$0.378790\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−94.2101	−3.08430
$934$	0	0
$935$	22.8484	0.747221
$936$	0	0
$937$	18.8253	0.614996	0.307498	−	0.951549i	$-0.400508\pi$
$937$	18.8253	0.614996	0.307498	+	0.951549i	$0.400508\pi$
$938$	0	0
$939$	−35.4747	−1.15767
$940$	0	0
$941$	13.8308	0.450871	0.225436	−	0.974258i	$-0.427619\pi$
$941$	13.8308	0.450871	0.225436	+	0.974258i	$0.427619\pi$
$942$	0	0
$943$	−15.4952	−0.504592
$944$	0	0
$945$	19.4952	0.634177
$946$	0	0
$947$	46.0105	1.49514	0.747570	−	0.664183i	$-0.231220\pi$
$947$	46.0105	1.49514	0.747570	+	0.664183i	$0.231220\pi$
$948$	0	0
$949$	93.6379	3.03961
$950$	0	0
$951$	−14.8536	−0.481660
$952$	0	0
$953$	7.44384	0.241130	0.120565	−	0.992705i	$-0.461529\pi$
$953$	7.44384	0.241130	0.120565	+	0.992705i	$0.461529\pi$
$954$	0	0
$955$	−6.30180	−0.203921
$956$	0	0
$957$	13.2043	0.426834
$958$	0	0
$959$	23.8416	0.769884
$960$	0	0
$961$	−29.8858	−0.964059
$962$	0	0
$963$	42.4951	1.36939
$964$	0	0
$965$	15.3789	0.495063
$966$	0	0
$967$	−41.6752	−1.34018	−0.670092	−	0.742278i	$-0.733745\pi$
$967$	−41.6752	−1.34018	−0.670092	+	0.742278i	$0.733745\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.2690	0.618372	0.309186	−	0.951002i	$-0.399943\pi$
$971$	19.2690	0.618372	0.309186	+	0.951002i	$0.399943\pi$
$972$	0	0
$973$	28.5506	0.915292
$974$	0	0
$975$	−21.2002	−0.678948
$976$	0	0
$977$	−19.7307	−0.631242	−0.315621	−	0.948885i	$-0.602213\pi$
$977$	−19.7307	−0.631242	−0.315621	+	0.948885i	$0.602213\pi$
$978$	0	0
$979$	11.6684	0.372924
$980$	0	0
$981$	−36.8344	−1.17603
$982$	0	0
$983$	−33.2636	−1.06094	−0.530472	−	0.847702i	$-0.677985\pi$
$983$	−33.2636	−1.06094	−0.530472	+	0.847702i	$0.677985\pi$
$984$	0	0
$985$	−12.7210	−0.405324
$986$	0	0
$987$	−54.5331	−1.73581
$988$	0	0
$989$	24.3434	0.774076
$990$	0	0
$991$	17.8161	0.565947	0.282973	−	0.959128i	$-0.408679\pi$
$991$	17.8161	0.565947	0.282973	+	0.959128i	$0.408679\pi$
$992$	0	0
$993$	92.1465	2.92418
$994$	0	0
$995$	10.2177	0.323923
$996$	0	0
$997$	21.2096	0.671715	0.335857	−	0.941913i	$-0.390974\pi$
$997$	21.2096	0.671715	0.335857	+	0.941913i	$0.390974\pi$
$998$	0	0
$999$	−137.291	−4.34371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7220.2.a.p.1.1		4
19.8	odd	6		380.2.i.c.121.1	✓	8
19.12	odd	6		380.2.i.c.201.1	yes	8
19.18	odd	2		7220.2.a.r.1.4		4
57.8	even	6		3420.2.t.w.1261.2		8
57.50	even	6		3420.2.t.w.3241.2		8
76.27	even	6		1520.2.q.m.881.4		8
76.31	even	6		1520.2.q.m.961.4		8
95.8	even	12		1900.2.s.d.349.8		16
95.12	even	12		1900.2.s.d.49.8		16
95.27	even	12		1900.2.s.d.349.1		16
95.69	odd	6		1900.2.i.d.201.4		8
95.84	odd	6		1900.2.i.d.501.4		8
95.88	even	12		1900.2.s.d.49.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.c.121.1	✓	8	19.8	odd	6
380.2.i.c.201.1	yes	8	19.12	odd	6
1520.2.q.m.881.4		8	76.27	even	6
1520.2.q.m.961.4		8	76.31	even	6
1900.2.i.d.201.4		8	95.69	odd	6
1900.2.i.d.501.4		8	95.84	odd	6
1900.2.s.d.49.1		16	95.88	even	12
1900.2.s.d.49.8		16	95.12	even	12
1900.2.s.d.349.1		16	95.27	even	12
1900.2.s.d.349.8		16	95.8	even	12
3420.2.t.w.1261.2		8	57.8	even	6
3420.2.t.w.3241.2		8	57.50	even	6
7220.2.a.p.1.1		4	1.1	even	1	trivial
7220.2.a.r.1.4		4	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 7220 = 2^{2} \cdot 5 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7220.a (trivial)

\(f(q)\)	\(=\)	\(q-3.16505 q^{3} +1.00000 q^{5} -1.53315 q^{7} +7.01755 q^{9} -3.79695 q^{11} +6.69820 q^{13} -3.16505 q^{15} -6.01755 q^{17} +4.85250 q^{21} +4.38565 q^{23} +1.00000 q^{25} -12.7158 q^{27} +1.09875 q^{29} +1.05555 q^{31} +12.0175 q^{33} -1.53315 q^{35} +10.7970 q^{37} -21.2002 q^{39} -3.53315 q^{41} +5.55070 q^{43} +7.01755 q^{45} -11.2381 q^{47} -4.64945 q^{49} +19.0459 q^{51} -1.57636 q^{53} -3.79695 q^{55} -3.23135 q^{59} +0.0663000 q^{61} -10.7590 q^{63} +6.69820 q^{65} +5.70500 q^{67} -13.8808 q^{69} +6.47760 q^{71} +13.9796 q^{73} -3.16505 q^{75} +5.82130 q^{77} +1.99320 q^{79} +19.1934 q^{81} -10.3477 q^{83} -6.01755 q^{85} -3.47760 q^{87} -3.07310 q^{89} -10.2693 q^{91} -3.34086 q^{93} -8.87005 q^{97} -26.6453 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} + 4 q^{5} + 5 q^{9} + 2 q^{11} + 9 q^{13} - q^{15} - q^{17} + 8 q^{21} + 4 q^{25} - 10 q^{27} + 5 q^{29} + 10 q^{31} + 25 q^{33} + 26 q^{37} - 27 q^{39} - 8 q^{41} - 7 q^{43} + 5 q^{45} - 16 q^{47}+ \cdots - 24 q^{99}+O(q^{100}) \) 4 * q - q^3 + 4 * q^5 + 5 * q^9 + 2 * q^11 + 9 * q^13 - q^15 - q^17 + 8 * q^21 + 4 * q^25 - 10 * q^27 + 5 * q^29 + 10 * q^31 + 25 * q^33 + 26 * q^37 - 27 * q^39 - 8 * q^41 - 7 * q^43 + 5 * q^45 - 16 * q^47 + 10 * q^49 + 12 * q^51 + 5 * q^53 + 2 * q^55 + 11 * q^59 - 12 * q^61 + 3 * q^63 + 9 * q^65 - 3 * q^69 + 14 * q^71 + 4 * q^73 - q^75 - 22 * q^77 + 13 * q^79 + 24 * q^81 + 5 * q^83 - q^85 - 2 * q^87 + 5 * q^89 - 46 * q^91 + 28 * q^93 - q^97 - 24 * q^99

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